Calculator Form
Example Data Table
| a | b | m | t | gcd(a, b) | Base (x, y) | Target | Selected (x(t), y(t)) | Check |
|---|---|---|---|---|---|---|---|---|
| 84 | 30 | 1 | 0 | 6 | (-1, 3) | 6 | (-1, 3) | 84(-1) + 30(3) = 6 |
| 84 | 30 | 4 | 0 | 6 | (-1, 3) | 24 | (-4, 12) | 84(-4) + 30(12) = 24 |
| 99 | 78 | 1 | 2 | 3 | (-11, 14) | 3 | (41, -52) | 99(41) + 78(-52) = 3 |
Formula Used
Bézout’s identity states that for integers a and b, there exist integers x and y such that:
ax + by = gcd(a, b)
The calculator uses the extended Euclidean algorithm to compute gcd(a, b) and one valid coefficient pair (x, y).
For a scaled target value m × gcd(a, b), one solution becomes:
x₀ = m × x
y₀ = m × y
All solutions for the target equation are generated by:
x = x₀ + (b / gcd(a, b))t
y = y₀ - (a / gcd(a, b))t
When both integers are non-zero, the least common multiple is:
lcm(a, b) = |ab| / gcd(a, b)
How to Use This Calculator
- Enter two integers in the a and b fields.
- Set multiplier m to choose a target of m × gcd(a, b).
- Set parameter t to inspect another valid family solution.
- Click Calculate to show the result below the header and above the form.
- Review gcd, coefficients, verification lines, and the general solution family.
- Use the CSV and PDF buttons to save the report.
- Read the Euclidean step table to study each division.
- Inspect the Plotly graph to see how the remainders decrease across iterations.
About This Bézout Theorem Calculator
This calculator is designed for integer coefficient problems based on Bézout’s identity. It gives more than a single gcd value. It also provides one exact coefficient pair, a scaled target solution, and a parametric family generated from the same identity. That makes it useful for both quick answers and step-by-step study.
The extended Euclidean algorithm sits at the center of the calculation. Each division step reduces the remainder until the gcd is reached. Once the gcd is known, the same backward structure produces coefficients that satisfy ax + by = gcd(a, b). The page then scales that solution by the multiplier m and adjusts it by parameter t to generate additional valid pairs.
The example table helps learners compare several inputs quickly. The verification lines show that every displayed solution really works. The graph adds a visual view of how the Euclidean process shrinks remainders through repeated division. Together, the table, equations, and graph make the output easier to trust and easier to explain in class, homework, or technical notes.
Because everything is kept in one page, the tool is easy to save, edit, and reuse. The export buttons create CSV and PDF reports from the computed result, making it practical for revision sheets, worksheets, or documentation. The layout stays simple while the calculator fields still adapt into three columns on large screens, two on smaller screens, and one on mobile devices.
FAQs
1. What does this calculator compute?
It computes gcd(a, b), one Bézout coefficient pair, a scaled target solution, a general solution family, Euclidean steps, and a remainder graph.
2. What is Bézout’s identity?
It states that integers x and y always exist such that ax + by equals gcd(a, b), provided a and b are not both zero.
3. Why is the multiplier m useful?
It scales the base coefficient pair so you can solve ax + by = m × gcd(a, b) instead of only the base gcd equation.
4. What does parameter t do?
It selects another valid solution from the infinite family. Different t values change x and y while preserving the same target value.
5. Can negative integers be used?
Yes. The calculator accepts positive, negative, and zero values. It adjusts coefficient signs correctly and still verifies the identity.
6. What happens when one value is zero?
The calculator still works if one input is zero. The non-zero value becomes the gcd magnitude, and valid coefficients are shown.
7. Why does the graph matter?
The graph visualizes remainder reduction during the Euclidean algorithm. It helps learners see convergence and understand the division sequence.
8. Why are there many correct coefficient pairs?
Once one solution exists, infinitely many others follow from the parametric family. Changing t shifts both coefficients without changing the target equation.